1. Sounds like you just dumped all the grains in the water to steep. Is this correct? In the future you can put the grains in a mesh bag. Or if it was because of the hops, you can buy pellet hops which will dissolve completely.

2. When you are ready for the hydrometer reading get a plastic thief, sanitize it, then draw out just enough to fill up a vessel large enough for the hydrometer (a graduated cylinder works well). This is much easier than trying to drop a hydrometer into a carboy and fish it out. It's also easier to read the hydrometer, and there is less risk of contamination.

1) When boiling, you create a lot of "hot break". It's normal. The best way to avoid clogging up your strainer is the whirlpool the kettle for a few minutes and let the break settle out to a cone in the center of the pot. Then siphon from the edge of the pot in to the carboy.

2) When starting to siphon from the pot/kettle in to the carboy, I run the first bit in to the hydrometer. I don't normally take any more hydrometer readings again until it goes in to the secondary.

1) You can also use as sanitized spoon to scoop the hops out of the strainer, if it becomes clogged.

2) I measure my gravity before my boil (make sure you get the volume, too), so that I can adjust the hops if necessary (I'm doing all grain, mostly). Then I get the volume in the pot, after the boil, and calculate my post-boil OG based on the pre-boil numbers.
During fermentation I use a thief.
At any transfers I get the gravity by catching the first bit from the siphon.

I did have the grains in a mesh bag and also used hop pellets, but still ended up with a bunch of gunk in the strainer. I ended up using a sanitized spoon to get it all out to continue the flow. Maybe next time, I'll try siphoning instead of pouring.